1. Temporal Extensions to Defeasible Logic
Guido Governatori1, Paolo Terenziani2
1 University of Quuensland, Brisbane, Australia
2Dipartimento di Informatica, UPO, Alessandria, Italy

2. Introduction Defeasible conclusions  nonmonotonic logic
Trade-off: expressiveness vs comp. complexity
Defeasible Logic [Nute,94]: a linear logic
Several applications:
- legal reasoning
- contracts and agent negotiations
- Semantic Web

3. Defeasible Logic Facts (predicate; e.g., penguin(Tweedy))
Strict Rules A1..An  B (classical rules)
Defeasible Rules A1..An B (rules that can be defeated by contrary evidence; e.g., “birds usually fly”)
Defeaters A1..An  B (rules to prevent derivation of conclusions; “e.g., if something is heavy it might not fly”)
Priorities between rules
“skeptical” nonmonotonic logic: it does not support contraddictory conclusions

4. Provability in DL Let D be a Theory
+q (q is definitely provable in D, i.e., using only facts and strict rules)
-q (we proved that q is not definitely provable in D)
+q (q is defeasibly provable in D)
- q (we proved that q is not defeasibly provable in D)

5. Derivability A conclusion p is derivable when p is a fact
there is an applicable strict or defeasible rule for p, and
- all the rules for  p are discarded (i.e., proved not to be applicable), or
- every applicable rule for  p is weaker than an applicable strict or defeasible ruple for p

6. Temporal Extensions
Explicit representation of time need to cope with large parts of reality (e.g., causation)
- durative actions
- delays
Trade-off between expressiveness and computational complexity
GOAL: temporal extension to DL retaining LINEAR complexity

7. Temporal Rules a1:d1, …., an:dn d b:db
e:d e is an event whose duration is exactly d (d1)
a1:d1, …., an:dn are the “causes”. They can start at different points in time
b:db is the effect
d is the exact delay between causes and effects

8. Temporal Rules a1:d1, …., an:dn d b:db SCHEMA OF RULES
d is the delay between the beginning of the last cause and the beginning of the effect
d is the delay between the ending point of the last cause and the beginning of the effect (here finite causes only)

9. TRIGGERING CONDITIONS (intuition):
Temporal Rules
TRIGGERING CONDITIONS (intuition):
We must be able to prove each ai for for exactly di consecutive time points, i.e., it0,t1,….,tdi, tdi+1 consecutive time points such that we can prove ai at points t1,….,tdi and we cannot prove it at t0 and tdi+1
Let tmax the last time when the latest cause can be proved
b can be proven for exactly db instants starting from time tmax+d

10. Example
F =
r1: a:1 10 d:10; r2: b:1 7 d:5; r3: c:1 8 d:5; r3  r2
0 … … …… a b c r2
+d
+d
r2 terminates r1
+d r1 r1
+d
r3  r2
+d
r3  r2
+d r3
+d

11. Proof Conditions for +@
If = P(n+1) then
+  P(1..n) or
(i)  P(1..n) and
(ii) rRsd[p] \ either r persists or r is -applicable at t and
(iii) sR[~p] either
- s is -discarded at t or
- if s is (t-t’)-effective,then vRsd[p]\ v defeats s at t’

12. Complexity
THEOREM 1
Let D be a temporalized defeasible theory without backward causation. Then the extension of D from time t0 to t (i.e., the set of all consequences of D derivable from t0 to t) can be computed in time linear to the size of the theory, i.e., O(|Prop||R|  t)

13. Causation a1:tad b:tb “Backward” causation: 0>ta+d
“One-shot” causation: 0ta+d and 0<tb+d
“Continuous” causation: 0ta+d and 0  tb+d
“Mutually sustaining” causation: 0=ta+d and 0 = tb+d
“Culminated event” causation: 0  d

14. Conclusions & Future Work
TEMPORAL EXTENSION TO DL
increased expressiveness
retaining linear complexity
FUTURE
Complexity of theories with backward causation
Type of events (e.g., states vs accomplishments vs processes)